Mathematics > Combinatorics

Title:
The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree

Abstract: It is known that every planar graph has a planar embedding where edges are
represented by non-crossing straight-line segments. We study the planar slope
number, i.e., the minimum number of distinct edge-slopes in such a drawing of a
planar graph with maximum degree $\Delta$. We show that the planar slope number
of every planar partial 3-tree and also every plane partial 3-tree is at most
$O(\Delta^5)$. In particular, we answer the question of Dujmovi\'c et al.
[Computational Geometry 38 (3), pp. 194--212 (2007)] whether there is a
function $f$ such that plane maximal outerplanar graphs can be drawn using at
most $f(\Delta)$ slopes.